Thank you for the question on Ishmael, “Why did God send Ishmael away in the Bible?”
God DIDN’T send Ishmael. It was Sarah who wanted Abraham to send Hagar and Ishmael away from her son, Isaac.
The Biblical text in Genesis 21 says this, “The child grew and was weaned, and on the day Isaac was weaned Abraham held a great feast. 9 But Sarah saw that the son whom Hagar the Egyptian had borne to Abraham was mocking, 10 and she said to Abraham, “Get rid of that slave woman and her son, for that woman’s son will never share in the inheritance with my son Isaac.” 11 The matter distressed Abraham greatly because it concerned his son. 12 But God said to him, “Do not be so distressed about the boy and your slave woman. Listen to whatever Sarah tells you, because it is through Isaac that your offspring will be reckoned. 13 I will make the son of the slave into a nation also, because he is your offspring” (A searchable online Bible in over 150 versions and 50 languages).
When Sarah became pregnant, sadly Ishmael was mocking Sarah and her newborn infant, Isaac, This angered Sarah that she wanted Hagar and her son sent away from her family. The text does not say how Ishmael mocked Sarah and her son, but the words were hurtful. Sarah believed that as long as Ishmael lived among their tents that will be a continual problem with their relationship. Her solution was to Abraham to send Hagar and Ishmael away. It was Abraham who conceded to her demand. God didn’t override Abraham and Sarah. He comforted Abraham with the promise that Ishmael will also be a nation. God didn’t reject or curse Ishmael since he was from Abraham offspring.
SUMMARY: Sarah wanted Abraham to send Hagar and Ishmael away for disrespecting her and her newborn infant. It was not God who sent Hagar and Ishmael away.
For more perspectives:
https://www.quora.com/Why-did-God-send-Ishmael-away-in-the-Bible